effects of chemical reaction on magneto micropolar fluid flow from a radiative surface with variable permeability

Int J of Applied Mechanics and Engineering, 2013, vol 18, No 3, pp 833 851 DOI 10 2478/ijame 2013 0050 EFFECTS OF CHEMICAL REACTION ON MAGNETO MICROPOLAR FLUID FLOW FROM A RADIATIVE SURFACE WITH VARIA[.]

EFFECTS OF CHEMICAL REACTION ON MAGNETO-MICROPOLAR FLUID FLOW FROM A RADIATIVE SURFACE WITH VARIABLE

PERMEABILITY

B.K SHARMA* Department of Mathematics Department of Civil Engineering Birla Institute of Technology and Science

Pilani Rajasthan, INDIA E-mail: bhupen_1402@yahoo.co.in

A.P SINGH Department of Civil Engineering Birla Institute of Technology and Science

Pilani Rajasthan, INDIA

K YADAV and R.C CHAUDHARY Department of Mathematics University of Rajasthan, Jaipur, INDIA

This paper presents a study of a hydromagnetic free convection flow of an electrically conducting micropolar

fluid past a vertical plate through a porous medium with a heat source, taking into account the homogeneous

chemical reaction of first order A uniform magnetic field has also been considered in the study which acts

perpendicular to the porous surface of the above plate The analysis has been done by assuming varying

permeability of the medium and the Rosseland approximation has been used to describe the radiative heat flux in

the energy equation Numerical results are presented graphically in the form of velocity, micro- rotation,

concentration and temperature profiles, the skin-friction coefficient, the couple stress coefficient, the rate of heat

and mass transfers at the wall for different material parameters The study clearly demonstrates how a chemical

reaction influences the above parameters under given conditions

Key words: chemical reaction, micro-polar fluid, micro-rotation, magneto-hydrodynamics, heat and mass

transfer

MSC 2000: 80A20, 76R10, 76S05

1 Introduction

In many engineering applications such as separation processes in chemical engineering, heat and mass transfer characteristics have been used widely For example, evaporation, condensation, distillation, rectification processes in fluids condensing or boiling at a solid surface play a decisive role (Baehr and Stephan, 1998) In order to study the theory of micropolar fluids, Eringen (1964) developed a simple theory which includes the effect of local rotary inertia, the couple stress and the inertial spin This theory is expected to be useful in analyzing the behavior of non-Newtonian fluids Eringen (1966) also developed the theory of micropolar fluids for the cases where only microrotational effects and microrotational inertia exist Eringen (1972) extended the theory of thermo-micropolar fluids and derived the constitutive law for fluids

with microstructure This general theory of micropolar fluids deviates from that of Newtonian fluids by

adding two new variables to the velocity These variables are micro-rotation that is spin and microinertia

tensor describing the distributions of atoms and molecules inside the microscopic fluid particles The theory

may be applied to explain the phenomenon of the flow of colloidal fluids, liquid crystals, polymeric

suspensions, animal blood etc In view of Lukaszewicz (1999), micropolar fluids represent those fluids

which consist of randomly oriented particles suspended in a viscous medium Several authors have studied

the characteristics of the boundary layer flow of a micropolar fluid under different boundary conditions An

excellent review of micropolar fluids and their applications was given by Ariman et al (1973) Gorla (1992)

also discussed the steady state heat transfer in a micropolar fluid flow over a semi-infinite plate Rees and

Pop (1998) studied free convection boundary layer flow of a micropolar fluid from a vertical flat plate

Takhar and Soundalgekar (1980; 1985) studied the flow and heat transfer of a micropolar fluid past a porous

plate Further, they (1983; 1985) discussed these problems for the flow past a continuously moving porous

plate A micropolar fluid flow over a horizontal plate with surface mass transfer was presented by Yucel

(1989) Gorla et al (1998; 1995) investigated further the concept of natural convection from a heated vertical

plate in a micropolar fluid

Flows of fluids through porous media are of principal interest because they are quite prevalent in

nature Such flows have attracted the attention of a number of scholars due to their applications in many

branches of science and technology, viz., in the fields of agriculture engineering to study the underground

water resources, seepage of water in river beds, in petroleum technology to study the movement of natural

gas, oil, and water through oil reservoirs, in chemical engineering for filtration and purification processes

Hiremath and Patil (1993) studied the effect of free convection currents on the oscillatory flow of a polar

fluid through a porous medium, which is bounded by a vertical plane surface of constant temperature

The problem of flow and heat transfer for a micropolar fluid past a porous plate embedded in a

porous medium has been of great interest in engineering studies such as oil exploration, thermal insulation,

etc Raptis and Takhar (1999) considered a micropolar fluid flow through a porous medium Fluctuating heat

and mass transfer of three-dimensional flow through a porous medium with variable permeability was

discussed by Sharma et al (2007)

Hydromagnetic convection with heat and mass transfer has been studied due to its importance in the

design of magnetohydrodynamics (MHD) generators and accelerators in geophysics, astrophysics, nuclear

power reactors and so on The interest in these new problems generates from their importance in liquid

metals, electrolytes and ionized gases The unsteady hydromagnetic free convection flow of Newtonian and

polar fluids was investigated by Helmy (1998) Chaudhary and Sharma (2006) considered combined heat and

mass transfer by laminar mixed convection flow from a vertical surface with induced magnetic field

Hydromagnetic unsteady mixed convection and mass transfer flow past a vertical porous plate immersed in a

porous medium was investigated by Sharma and Chaudhary (2008) El-Hakien et al (1999) studied the

effects of viscous flow and Joule heating on the MHD-free convection flow with variable plate temperature

in a micropolar fluid El-Amin (2001) considered the MHD free-convection and mass transfer flow in a

micropolar fluid over a stationary vertical plate with constant suction Kim (2001) investigated the unsteady

free convection flow of a micropolar fluid past a vertical plate embedded in a porous medium and extended

his work (2004) to study the effects of heat and mass transfer in the MHD micropolar fluid flow past a

vertical moving plate Analytical studies on the MHD flow of a micropolar fluid over a vertical porous plate

were presented by Kim and Lee (2003) and Helmy et al (2002)

Combined heat and mass transfer problems with chemical reaction are of importance in many

processes and have, therefore, received a considerable amount of attention in recent years In processes such

as drying, evaporation at the surface of water body, energy transfer in wet cooling tower and the flow in a

desert cooler, heat and mass transfer occur simultaneously Chemical reactions can be codified as either

homogeneous or heterogeneous processes A homogeneous reaction is one that occurs uniformly through a

given phase In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of

a phase A reaction is said to be of first order, if the rate of reaction is directly proportional to the

concentration itself which has many applications in different chemical engineering processes and other

industrial applications such as polymer production, manufacturing of ceramics or glassware and food

processing (Cussler, 1998) Das et al (1994) considered the effects of first order chemical reaction on the

flow past an impulsively started infinite vertical plate with constant heat flux and mass transfer Muthucumarswamy and Ganesan (2001) and Muthucumarswamy (2002) studied a first order homogeneous chemical reaction on flow past an infinite vertical plate

In the above mentioned studies the effects of heat sources/sinks and radiation have not been considered Many processes in new engineering areas occur at high temperature and knowledge of heat transfer becomes imperative for the design of the pertinent equipment Nuclear power plants, gas turbines and various propulsion devices for aircraft, missiles, satellites, and space vehicles are examples of such

engineering areas Kandasamy et al (2005) discussed the heat and mass transfer effect along a wedge with

heat source and concentration in the presence of suction/injection taking into account the chemical reaction

of first order Sharma et al (2006a; 2006b) reported on the radiation effect with simultaneous thermal and

mass diffusion in the MHD mixed convection flow from a vertical surface Perdikis and Repatis (1996) illustrated the heat transfer of a micropolar fluid in the presence of radiation Raptis (1998) studied the effect

of radiation on the flow of a micropolar fluid past a continuously moving plate Elbashbeshby and Bazid (2000) and Kim and Fedorov (2003) reported on the radiation effects on the mixed convection flow of a micropolar fluid Moreover, when the radiative heat transfer takes place, the fluid involved can be electrically conducting in the sense that it is ionized owing to high operating temperature The heat and mass transfer in a magneto hydrodynamic micropolar fluid flow through a porous medium under different physical

conditions was examined by Ibrahim et al (2004) Since the flow past a continuously moving plate has many

applications in manufacturing processes, Rahman and Sattar (2006) analyzed the MHD convective flow of a micropolar fluid past a continuously moving vertical porous plate in the presence of heat generation/absorption The process of fusing of metals in an electrical furnace by applying a magnetic field and the process of cooling of the first wall inside a nuclear reactor containment vessel where the hot plasma

is isolated from the wall by applying a magnetic field are examples of such fields where thermal radiation and magneto-hydrodynamics are correlative This fact was taken into consideration by Aziz (2006) in his

study on micropolar fluids Sharma et al (2007; 2008) discussed the radiation effect on the free convective

flow along a uniform moving porous vertical plate in the presence of heat source/sink and transverse magnetic field Recently, Chaudhary and Jain (2007) investigated the combined heat and mass transfer in a magneto-micropolar fluid flow from a radiative surface with variable permeability In the present analysis, it

is proposed to study the effect of a first order homogeneous chemical reaction on a magneto-micropolar fluid flow past a radiative vertical plate through a porous medium with a heat source

2 Mathematical formulation

In this study, the two-dimensional flow of a micropolar fluid past a semi-infinite vertical plate

embedded in a porous medium has been considered by taking the x-axis along the porous plate and the y-axis

normal to it as shown in Fig.1 Due to the semi-infinite plane surface assumption, the flow variables are

functions of y and t only In order to derive the basic equations for the problem under investigation, the

following assumptions have been made:

1 The fluid under consideration is viscous, incompressible and electrically conducting with constant

physical properties

2 The flow is unsteady, laminar and the magnetic field is applied perpendicular to the plate

3 Viscous and Darcy’s resistance terms are taken into account with variable permeability porous medium

4 The magnetic Reynolds number is assumed to be small enough so that the induced magnetic field can be

neglected

5 The Hall Effect, Joule Heating, and viscous dissipation are all neglected in this study

6 The fluid is considered to be a gray, absorbing-emitting but non-scattering medium and the Rosseland

approximation is used to describe the radiative heat flux

7 There is a first order chemical reaction between the diffusing species and the fluid

8 It is also assumed that there is no applied voltage which implies the absence of an electric field

Under these assumptions and introducing non-dimensional quantities, the governing equations for the

flow are as follows

+

2 nt

nt

(2.1)

2

3 Pr

2

2 nt

1

where

*

,

P

M

with the corresponding boundary conditions

y

w

3 Solution for the problem

In order to reduce the aforesaid system of partial differential equations to a system of ordinary

differential equations in a dimensionless form, we may write the translational velocity, microrotation,

temperature and concentration as

ntt 2 ,

nt 2 ,

ntt 2 ,

By substituting the above Eqs (3.1) – (3.4) into Eqs (2.1) – (2.4), neglecting the coefficient of 0(2),

we obtain the following pairs of equations for (u 0, Z0, T0 , C 0 ) and (u 1, Z1, T1 , C 1)

2

Au

1 Bv u cc uc Mn u  T  C 2BvZ c Buc

,

0cc 0c 0

,

1cc 1c n 1 B 0c

,

0 b 1 0 Sb 1 0 0

1 b 1 1 b S 1 n 1 Bb 1 0

where the primes denote differentiation with respect to y



The corresponding boundary conditions can be written as

, , , , ,

The solutions of Eqs (3.6) – (3.12) with the satisfying boundary conditions (3.13) and (3.14) are

given by

,

F y 1 ,

0 y L e 6 

F 3 y F y 1 ,

1 y L e 10  L e 9 

a y 2 ,

0 y e

a y 2 a y 2 ,

1 y c e 2  c e 3 

a y 3 ,

0

a y 4 a y 3 a y 2 a y 1

The constants are not given for the sake of brevity

4 Results and discussion

The MHD convective flow and mass transfer of an incompressible micropolar fluid along a

semi-infinite vertical plate in a porous medium with a heat source and chemical reaction has been studied in

preceding sections In order to get a physical insight into the problem, numerical calculations for the

distribution of the translational velocity, microrotation, temperature and concentration rate of heat transfer

and mass transfer across the boundary layer for various values of the parameter have been done The effects

of the main controlling parameters as they appear in the governing equations are discussed in the current

section In this study, entire numerical calculations have been performed with  = 0.01, n = 1 and t = 1 while

Pr, R, Sc, Gr, Gc, Bv, M, , S, m, B, K 1 and A are varied over ranges, which are listed in the figure legends In

the absence of the chemical reaction and heat source, temperature profiles and concentration profiles have

been analyzed they are in good agreement with the available results of Chaudhary and Jain (2007) as shown

in Figs 2 and 4, respectively

Typical variations of the temperature profiles (y) along the span-wise coordinate y are shown in Fig

3 for different values of the Prandtl number (Pr =0.71, for air at 20 0 C and 1 atmospheric pressure, Pr =1.0 for electrolytic solution, at 20 0 C and 1 atmospheric pressure), the radiation parameter (R) and suction parameter (B) The numerical results show that the temperature decreases with an increase in the Prandtl number This

is due to the fact that a fluid with a high Prandtl number has a relatively low thermal conductivity which results in the reduction of the thermal boundary layer thickness Also, the figure indicates that the

temperature reduces with an increase in the radiation parameter (R), suction parameter (B) and heat source

for air while a reverse effect is observed for the electrolytic solution

Figure 4 depicts the species concentration for different gases The values of the Schmidt number

(Sc) are chosen to represent the most common diffusing chemical species like hydrogen (Sc = 0.22), oxygen (Sc = 0.66), and ammonia (Sc = 0.78) at a temperature of 25 0 C and 1 atmospheric pressure For Sc=0.22, B=0.1 and K1=0, the concentration profile is same as obtained by Chaudhary and Jain (2007) A comparison of curves in the figure shows a decrease in concentration distribution C(y) with an increase in

the Schmidt number because the smaller values of Sc are equivalent to increasing chemical molecular

diffusivity (D) Hence the concentration of the species is higher for small values of Sc and lower for larger values of Sc The concentration profiles also decrease with an increase in the suction parameter (B) There

is a fall in the concentration due to increasing values of the chemical reaction parameter This shows that the diffusion rates can be significantly altered by a chemical reaction Both the temperature and the

concentration profiles attain their maximum values at the wall and decrease exponentially with y and finally tend to zero as y

Fig.2 Temperature profiles for B=0.1, R=1.0 and S=0

Fig.3 Temperature profiles for =0.01, n=0.1, t=1

Fig.4 Concentration profiles for =0.01, n=0.1, t=1

For different values of the radiation parameter (R), chemical reaction parameter (K 1), viscosity ratio

parameter (B v ), permeability parameter (), the translational velocity u and microrotation profiles  are plotted

in Figs 5 and 6, respectively It is noteworthy that the velocity u and the magnitude of angular velocity  decrease as the radiation parameter (R) increases This result can be explained by the fact that a decrease in the radiation parameter R = ka R /4 *T 3 ’ for a given k and Tf, means a decrease in the Rosseland radiation

absorbtivity (a R) In view of Eqs (A5) and (A6), it is concluded that the divergence of the radiative heat flux

q r /y* increases as a R decreases and this means that the rate of radiative heat transferred to the fluid increases

and consequently the fluid temperature (see Fig.3) and hence the velocity of its particles also increases

Moreover, figures reveal that on increasing the values of the permeability parameter () the profiles of u and

the magnitude of , across the boundary layer, tend to increase It is noted that translational velocity increases with decreasing the chemical reaction parameter, while, a reverse effect is observed for microrotation profile

The velocity distribution decreases with increasing B v The phenomenon reflects the the fact that the effect of

increase in the value of B v will result in an enhancement of the total viscosity in the fluid flow because B v is

directly proportional to vortex viscosity which makes the fluid flow more viscous and so weakens the

convection currents In addition, the magnitude of  increases as B v increases

Fig.5 Velocity profiles u for Pr=1, M=1, Gr=2, S=1, m=0.5, Sc=0.22, B=0.5

Fig.6 Microrotation profiles  for Pr=1, M=1, Gr=2, S=1, m=0.5, Sc=0.22, B=0.5

Figures 7 and 8 illustrate the translational velocity and microrotation profiles against the span-wise

coordinate y for various values of the Prandtl number (Pr), Grashof number (Gr), magnetic parameter (M),

heat source parameter (S), m and Schmidt number (Sc), respectively It is observed that, keeping other

parameters fixed, as the magnetic parameter increases, the translational velocity u decreases The presence

of a magnetic field in an electrically conducting fluid introduces a force called Lorentz force which acts

against the flow if the magnetic field is applied in the normal direction as considered in the present

problem This type of resistive force tends to slow down the flow field It is clear that there exists an

overshooting of the velocity u for small values of M (e.g.; M = 1.0) Since the magnetic field has a

stabilizing effect, the velocity overshoot decreases with increasing M and vanishes for higher values of M

(e.g.; M = 3) It is noticed that an increase in Gr leads to a rise in the values of velocity u because the

favorable buoyancy force accelerates the fluid, while a reverse effect is noted for the magnitude of

microrotation component  Furthermore, it is seen that the effect of increasing values of Sc results in

decreasing u, while an opposite behavior is observed for microrotation  across the boundary layer It is

reported that as the coefficient m increases, the translational velocity and the magnitude of microrotation

profiles increase Furthermore, as expected, the translational velocity and the magnitude of microrotation

at the wall decrease on increasing the values of Pr and S

distribution of the translational velocity, microrotation, temperature and concentration rate of heat transfer

and mass transfer across the boundary layer for various values of the parameter have...

the absence of the chemical reaction and heat source, temperature profiles and concentration profiles have

been analyzed they are in good agreement with the available results of Chaudhary... rates can be significantly altered by a chemical reaction Both the temperature and the

concentration profiles attain their maximum values at the wall and decrease exponentially with y and